The two were all-square through 15, but Clagett won No. 16 with a bogey to move 1-up. At the 17th, Clagett landed in a greenside bunker, but blasted out to three feet. Lyford lagged his 45-foot birdie try to 4 feet, but missed that putt. Clagett rolled in his par putt and won the match.
'That's probably one of the best bunker shots I can remember,' said Clagett, who established a new 36-hole stroke play record with a 134 total. 'It was uphill all the way, you couldn't even see the top of my head.'
Second-seeded Alan Foster, who is the reigning British Amateur champion, also moved on to the second round of match play. He bested Don Kuehn, 3 and 2.
Three former champions also moved on to Tuesday's second round.
Defending champion Mark Bemowski dispatched Frank Saxon, 3 and 2, and is relieved that one of the hardest parts appears to be over.
'One at a time. The first one is always the toughest,' said Bemowski.
Greg Reynolds, the 2002 winner, handled 1992 runner-up Bob Hullender, 4 and 3.
Kemp Richardson, who titled at this event in 2001 and 2003, bounced A.B. Sisco by a 4 and 3 margin.
The second and third rounds will take place on Tuesday, followed by the quarterfinals and semifinals on Wednesday. The final round is scheduled for Thursday.